Convex Geometry

Department: 
MATH
Course Number: 
8803-BLE
Hours - Lecture: 
3
Hours - Total Credit: 
3
Typical Scheduling: 
Not Regularly Scheduled

In short, I plan to cover major parts of chapters I, II, IV before covering some special topics (time permitting). Possible special topics include polytopes, lattice point enumeration (with applications to integer programming), applications in semidefinite optimization and moment problems in analysis. Polytopes and lattice point enumeration can be taught from the main textbook (also potentially using Ziegler’s textbook), while applications in semidefinite optimization and moment problems will require the use of “Semidefinite Optimization and Convex Algebraic Geometry” and some research articles.

Prerequisites: 

The course should be accessible to first or second year graduate students. The students are required to have thorough knowledge and understanding of linear algebra.
 

Course Text: 

“A course in Convexity”, A. Barvinok, “Lectures on Polytopes” – G. Ziegler, “Semidefinite Optimization and Convex Algebraic Geometry” – G. Blekherman, P. Parrilo, R.Thomas, editors

Topic Outline: 

• Definitions. Radon’s theorem. Helly’s Theorem. Carath´eordory’s Theorem.
• Polyhedra. Fourier-Motzkin elimination.
• Boundary Structure of convex sets. Separation Theorems. Extreme points. Krein-Milman
Theorem. Convex Cones.
• Applications to combinatorial polytopes. Applications to univariate moment problems and
numerical integration.
• Cone of positive semidefinite matrices. Cone of nonnegative polynomials and applications.
• Polarity, duality.
• Introduction to linear programming, dual problems, duality gap.
• Applications in semidefinite optimization.
• Applications in multivariate truncated moment problems.